nLab first uncountable ordinal




The category of countable ordinals and simulations, ordered by inclusion, is a preorder and in fact equivalent to a woset. In material set theory, this woset is often identified with the first uncountable ordinal (well-ordered set under the membership relation), often denoted ω 1\omega_1. Structurally, we can define ω 1\omega_1 up to isomorphism as a skeleton of the category of countable ordinals, or the Hartogs number of a countably infinite set, and so the study of countable ordinals becomes a study of the order structure of ω 1\omega_1.

This ω 1\omega_1 has no top element (of course not: otherwise ω 1\omega_1 would be a countable ordinal), and much of the interest and fun of the subject lies in playing the childhood game of seeing who can name the bigger number, or in this case the bigger countable ordinal. A key fact is that ω 1\omega_1 is countably cocomplete, and therefore behaves like a self-contained universe with respect to countably cocontinuous operations. There is quite a lot of scope in what one can build up to using such operations. After playing the larger ordinal game for a while, and becoming impressed by the sizes of ordinals one can define – while recognizing at the same time that one is “never getting anywhere” relative to ω 1\omega_1 itself – it is hard not to become awestruck by the staggering immensity of ω 1\omega_1.

On the other hand, the cardinality of ω 1\omega_1 is of course no bigger than that of, say, the real numbers \mathbb{R} (and the same as it if the continuum hypothesis holds).


See also

Last revised on February 21, 2024 at 07:21:04. See the history of this page for a list of all contributions to it.